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Analysis of a prototypicalmultiscale method coupling atomistic and continuum mechanics

Published online by Cambridge University Press:  15 August 2005

Xavier Blanc ,
Claude Le Bris  and
Frédéric Legoll
Xavier Blanc
Affiliation:
Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris, France. [email protected]
Claude Le Bris
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. [email protected]; [email protected]
Frédéric Legoll
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. [email protected]; [email protected]
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Abstract

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

Keywords

Multiscale methodsvariational problemscontinuum mechanicsdiscrete mechanics.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 797 - 826
Copyright
© EDP Sciences, SMAI, 2005

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References

Alberti, G. and Mantegazza, C., A note on the theory of SBV functions. Bollettino U.M.I. Sez. B 7 (1997) 375–382.
L. Ambrosio, L. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (2000).
X. Blanc, C. Le Bris and F. Legoll, work in preparation, and Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, Preprint, Laboratoire Jacques-Louis Lions, Université Paris 6 (2004), available at http://www.ann.jussieu.fr/publications/2004/R04029.html
Blanc, X., Le Bris, C. and Lions, P.-L., From molecular models to continuum mechanics. Arch. Rational Mech. Anal. 164 (2002) 341–381. CrossRef
Braides, A., Dal Maso, G. and Garroni, A., Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal. 146 (1999) 23–58. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1991).
Broughton, J.Q., Abraham, F.F., Bernstein, N. and Kaxiras, E., Concurrent coupling of length scales: Methodology and application. Phys. Rev. B 60 (1999) 2391–2403. CrossRef
Ciarlet, P.G., An O(h²) method for a non-smooth boundary value problem. Aequationes Math. 2 (1968) 39–49. CrossRef
P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, North Holland (1988).
P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1991) 17–351.
W. E and P. Ming, private communication.
Knap, J. and Ortiz, M., An analysis of the Quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899–1923. CrossRef
P. Le Tallec, Numerical Methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. III, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1994) 465–622.
F. Legoll, Méthodes moléculaires et multi-échelles pour la simulation numérique des matériaux (Molecular and multiscale methods for the numerical simulation of materials), Ph.D. Thesis, Université Pierre et Marie Curie (France), 2004, available at http://cermics.enpc.fr/~legoll/these_Legoll.pdf
J.E. Marsden and T.J.R. Hugues, Mathematical foundations of Elasticity. Dover (1994).
Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M., Quasicontinuum simulation of fracture at the atomic scale. Model. Simul. Mater. Sci. Eng. 6 (1998) 607–638. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997).
Tadmor, E.B., Ortiz, M. and Phillips, R., Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529–1563. CrossRef
Tadmor, E.B. and Phillips, R., Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529–4534. CrossRef
Tadmor, E.B., Smith, G.S., Bernstein, N. and Kaxiras, E., Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59 (1999) 235–245. CrossRef
Shenoy, V.B., Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M., Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80 (1998) 742–745. CrossRef
V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptative finite element approach to atomic-scale mechanics – the Quasicontinuum method, J. Mech. Phys. Solids 47 (1999) 611–642. CrossRef
C. Truesdell and W. Noll, The nonlinear field theories of mechanics theory of elasticity. Handbuch der Physik, III/3, Springer Berlin (1965) 1–602.
L. Truskinovsky, Fracture as a phase transformation, in Contemporary research in mechanics and mathematics of materials, Ericksen's Symposium, R. Batra and M. Beatty, Eds., CIMNE, Barcelona (1996) 322–332.
Van Vliet, K.J., Li, J., Zhu, T., Yip, S. and Suresh, S., Quantifying the early stages of plasticity through nanoscale experiments and simulations. Phys. Rev. B 67 (2003) 104105. CrossRef
Zhang, P., Klein, P.A., Huang, Y., Gao, H. and Numerical, P.D. Wu simulation of cohesive fracture by the virtual-internal-bond model. Comput. Model. Engrg. Sci. 3 (2002) 263–289.